Soft Questions,Hard Answers: Jacob Bernoulli's Probability in Historical Context

On the 300th anniversary of the publication of Jacob Bernoulli's Ars Conjectandi, the book is revisited in historical context and seen in a new light. Bernoulli's announced goal of extending the application of probability from games to civil life was frustrated by the a priori difficulty of specifying equally likely cases. The great approximation theorem Bernoulli proved and suggested could lead to empirical applications was impractical for all but unrealistically large sample sizes, but it was immensely influential in the development of the mathematics of probability.     

Reading Keynes' Treatise on Probability

A Treatise on Probability was published by John Maynard Keynes in 1921. The Treatise contains a critical assessment of the philosophical foundations of probability and of the statistical methodology at the time. We review the aspects of the book that are most related with statistics, avoiding uninteresting neophyte's forrays into philosophical issues. In particular, we examine the arguments provided by Keynes against the Bayesian approach, as well as the sketchy alternative of a return to Lexis' theory of analogies he proposes. Our conclusion is that the Treatise is a scholarly piece of work looking at past advances rather than producing directions for the future.     

Probability Integrals of the Multivariate t Distribution

Results on probability integrals of multivariate t distributions are reviewed. The results discussed include: Dunnett and Sobel's probability integrals, Gupta and Sobel's probability integrals, John's probability integrals, Amos and Bulgren's probability integrals, Steffens' non‐central probabilities, Dutt's probability integrals, Amos' probability integral, Fujikoshi's probability integrals, probabilities of cone, probabilities of convex polyhedra, probabilities of linear inequalities, maximum probability content, and Monte Carlo evaluation.     

Market Structure and Racial and Gender Discrimination:

The Annual Equal Employment Opportunity Reports and Current Population Survey files are used to examine employment and earnings by race and gender in the telecommunications industry following changes in the industry's affirmative action guidelines and following the divestiture of the American Telephone and Telegraph Company. Nonblack minority men were the only underrepresented group to experience marked employment gains as managers and professional. The relative employment probability of other underrepresented workers remained similar to the levels experienced during the implementation of affirmative action guidelines. Earnings differentials did decline for minority men and white women in the telecommunications industry. However, this finding is only unique to black men as the earnings pattern for white women and nonblack minority men mirror that found for these groups nationally.     

Developing a Reputation for Reticence

A sender who has disclosable information with probability less than one may partially conceal bad news by choosing to withhold information and pooling with uninformed types. The success of this strategy depends on receivers' beliefs about the probability that the sender has disclosable news. In a dynamic context, informed senders try to cultivate a reputation for reticence either by concealing good news along with the bad, or by concealing some good news and disclosing some bad news. A reputation for reticence is valuable because it makes receivers less skeptical of past or future nondisclosures. The model provides insight into the choice by firms such as Google not to disclose quarterly earnings guidance to analysts, as well as Tony Blair's reticence over his son's vaccine record during the measles–mumps–rubella scare in the United Kingdom.     

Forensic identification: the island problem and its generalisations

In forensics it is a classical problem to determine, when a suspect S shares a property Γ with a criminal C, the probability that S = C. In this article we give a detailed account of this problem in various degrees of generality. We start with the classical case where the probability of having Γ, as well as the a priori probability of being the criminal, is the same for all individuals. We then generalize the solution to deal with heterogeneous populations, biased search procedures for the suspect, Γ‐correlations, uncertainty about the subpopulation of the criminal and the suspect, and uncertainty about theΓ‐frequencies. We also consider the effect of the way the search for S is conducted, in particular when this is done by a database search. A returning theme is that we show that conditioning is of importance when one wants to quantify the ‘weight’ of the evidence by a likelihood ratio. Apart from these mathematical issues, we also discuss the practical problems in applying these issues to the legal process. The posterior probabilities of C = S are typically the same for all reasonable choices of the hypotheses, but this is not the whole story. The legal process might force one to dismiss certain hypotheses, for instance when the relevant likelihood ratio depends on prior probabilities. We discuss this and related issues as well. As such, the article is relevant both from a theoretical and from an applied point of view.     

Fisher's The Nature of Capital and Income*

Abstract . In this essay James Tobin reminds us of the importance of Irving Fisher's great book and how that book was favorably received by economists. Fisher presented fundamental definitions and supplied a path breaking exposition of the mathematics of interest rates and their relationship to the valuation of assets. This essay was originally published as the “Editorial Introduction to Volume 2 of The Works of Irving Fisher, 14 vols London: Pickering & Chatto, 1997; 2:1–6.     

Clarence E. Ayres and the Legacy of German Idealism

Abstract . Based on the Clarence E. Ayres (1891-1972) papers at the University of Texas, this study traces the continuity among thinkers such as Kant, Hegel, Veblen, Dewey, and Ayres; the latter was a leading institutional economist following Veblen's death. Publicly acknowledging his intellectual debt to Veblen and Dewey, Ayres drew from these men some idealistic assumptions as well as the historicism that is implicit in his technological determinism or instrumental theory of knowledge. Thorstein Veblen and John Dewey owed a great deal to the philosophical tradition of idealism, regardless of the devotion to naturalism in their systems. The origins of Ayres's technological theory of value are found in Veblen and Dewey writings and back of them the legacy of German idealism. The vital link was a mutual acceptance that freedom was expressed in a cultural and historical form, realized in human activity. It was a process.     

Some applications of the fast Fourier transform algorithm in insurance mathematics This paper is dedicated to Professor W. S. Jewell on the occasion of his 60th birthday

Transform methods, together with the fast Fourier transform algorithm, can be used to compute various quantities of interest in risk theory and insurance mathematics. These include the total claim size distribution at a fixed time, the mean and variance of the claim size process as a function of time in the Sparre-Andersen model, and the probability of ruin. The associated discretization error can be reduced by applying Richardson's deferred approach to the limit. A theorem is given that puts the use of this technique on a mathematical basis in the context of compound distributions.     

TOWARDS AN ALGORITHMIC REVOLUTION IN ECONOMIC THEORY

Abstract This is an attempt to tell a coherent story of a possible path towards an algorithmic revolution in economic theory, based on foundational debates in mathematics. First, by exposing the non‐computational content of classical mathematics, and its foundations, it is shown that both set theory and the tertium non datur can be dispensed with, as foundational concepts. Next, then, as a natural sequel, it follows that every kind of economic theory that bases its theoretical underpinning on classical mathematics can be freed from these foundations and can be made naturally algorithmic. This will make the subject face, in an all pervasive way, absolutely (algorithmically) undecidable decision problems. The thrust of the path towards an algorithmic revolution in economics lies, then, in pointing out that only a radically new mathematical vision of microeconomics, macroeconomics, behavioural economics, game theory, dynamical systems theory and probability theory can lead us towards making economic theory a meaningfully applied science and free of mysticism and subjectivism.     

高等数学在经济中的应用

高等数学在经济研究中起着基础性作用,只有学好高等数学才能更好的理解剖析经济现象掌握经济知识。本文主要用数学分析、常微分方程、高等代数、概率与数理统计等课程的相关知识来说明高等数学在经济中的应用。     

Jefferson's Unheavenly City: A Bicentennial Look*

Abstract . The opportunity provided by the American Bicentennial for a re-examination of our political values is also an opportunity for a closer look at the political philosophy of Thomas Jefferson. Although Jefferson has been placed, with John Locke, in a “heavenly city of the eighteenth century philosophers” who sought new defenses on behalf of medieval spiritualism and divine law, the present essay contends that Jefferson's epistemological commitments differed from Locke's, and that Jefferson's political theory was far more “modern” than Locke's with respect to the key notions of rights, property, and consent. Some of Jefferson's political conclusions differed from those of Locke either in degree, such as in the details of representation, or in kind, such as in regard to the ownership of property.     

On the asymptotic behaviour of the Stringer bound1

The Stringer bound is a widely used nonparametric 100(1 -α)% upper confidence bound for the fraction of errors in an accounting population. This bound has been found in practice to be rather conservative, but no rigorous mathematical proof of the correctness of the Stringer bound as an upper confidence bound is known and also no counterexamples are available. In a pioneering paper Bickel (1992) has given some fixed sample support to the bound's conservatism together with an asymptotic expansion in probability of the Stringer bound, which has led to his claim of the asymptotic conservatism of the Stringer bound. In the present paper we obtain expansions of arbitrary order of the coefficients in the Stringer bound. As a consequence we obtain Bickel's asymptotic expansion with probability 1 and we show that the asymptotic conservatism holds for confidence levels 1 -α, with α∈ (0,1/2]. It means that in general also in a finite sampling situation the Stringer bound does not necessarily have the right confidence level. Based on our expansions we propose a modified Stringer bound which has asymptotically precisely the right nominal confidence level. Finally, we discuss other consequences of the expansions of the Stringer bound such as a central limit theorem, a law of the iterated logarithm and the functional versions of them.     

The Monty Hall problem is not a probability puzzle* (It's a challenge in mathematical modelling)

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice? The answer is ‘yes’ but the literature offers many reasons why this is the correct answer. This article argues that the most common reasoning found in introductory statistics texts, depending on making a number of ‘obvious’ or ‘natural’ assumptions and then computing a conditional probability, is a classical example of solution driven science. The best reason to switch is to be found in von Neumann's minimax theorem from game theory, rather than in Bayes’ theorem.     

Why Distinguish Between Statistics and Mathematical Statistics–The Case of Swedish Academia

A separation between the academic subjects statistics and mathematical statistics has existed in Sweden almost as long as there have been statistics professors. The same distinction has not been maintained in other countries. Why has it been kept for so long in Sweden, and what consequences may it have had? In May 2015, it was 100 years since Mathematical Statistics was formally established as an academic discipline at a Swedish university where Statistics had existed since the turn of the century. We give an account of the debate in Lund and elsewhere about this division during the first decades after 1900 and present two of its leading personalities. The Lund University astronomer (and mathematical statistician) C. V. L. Charlier was a leading proponent for a position in mathematical statistics at the university. Charlier's adversary in the debate was Pontus Fahlbeck, professor in political science and statistics, who reserved the word statistics for ‘statistics as a social science’. Charlier not only secured the first academic position in Sweden in mathematical statistics for his former PhD student Sven Wicksell but also demonstrated that a mathematical statistician can be influential in matters of state, finance as well as in different natural sciences. Fahlbeck saw mathematical statistics as a set of tools that sometimes could be useful in his brand of statistics. After a summary of the organisational, educational and scientific growth of the statistical sciences in Sweden that has taken place during the last 50 years, we discuss what effects the Charlier–Fahlbeck divergence might have had on this development.     

Coherent forecasting for count time series using Box–Jenkins's AR(p) model

During the last three decades, integer‐valued autoregressive process of order p [or INAR(p)] based on different operators have been proposed as a natural, intuitive and maybe efficient model for integer‐valued time‐series data. However, this literature is surprisingly mute on the usefulness of the standard AR(p) process, which is otherwise meant for continuous‐valued time‐series data. In this paper, we attempt to explore the usefulness of the standard AR(p) model for obtaining coherent forecasting from integer‐valued time series. First, some advantages of this standard Box–Jenkins's type AR(p) process are discussed. We then carry out our some simulation experiments, which show the adequacy of the proposed method over the available alternatives. Our simulation results indicate that even when samples are generated from INAR(p) process, Box–Jenkins's model performs as good as the INAR(p) processes especially with respect to mean forecast. Two real data sets have been employed to study the expediency of the standard AR(p) model for integer‐valued time‐series data.     

Henry George and Europe

Abstract . Henry George's land reform ideas became known in Germany not through his writing or speaking but through the efforts of Michael Flürscheim, an industrialist and pioneer social reformer, who first presented those ideas to the public. The American's idea that the land value tax was the only legitimate source of government revenue as the only economic surplus had found no acceptance among German socialist leaders. It was a capitalist, Flürscheim, who was inspired by George's theories and wrote and spoke about them. Flürscheim brought about the foundation of the first German land reform organization. Though it failed, a successor became the largest such association in the world.     

Henry George and Europe: As Dissident Economist and Path-breaking Philosopher, He was a Catalyst for British Social Reform

Abstract . Henry George's influence was greater in the United Kingdom than in the United States. The 80s and 90s there were particularly favorable for the reception of his revolutionary ideas. Though, thanks to such thinkers as Alfred Russell Wallace and James and John Stuart Mill, a land reform movement already existed, its sudden rise to national significance was due to George. George's writing and speaking skills and his dedication moved many serious citizens into the political Left and heavily influenced men and women who became leaders of British non-Marxian socialism, at the formation and consolidation of their movement. While George's followers broke with both the Wallace and socialist movements, George's rhetorical talents awakened the broad circles of thinking people to a consciousness of the full range of the social question.     

Developing a brand from within: Involving employees and volunteers when developing a new brand position

In February 1996 Barnardo's invited six of the UK's leading advertising agencies to pitch for the provision of full and above the line communications and marketing services. Banner McBride was included as part of a joint WPP communications and marketing team (including Ogilvy & Mather) selected to support Barnardo's efforts to reposition and communicate the Barnardo's brand. As the WPP team said at the beginning, Barnardo's were ‘a well-known brand that was not known well’. The challenge was to increase the organisation's saliency, but at the same time build on years of heritage and ensure that employees and volunteers were included from the beginning. This paper describes how the brand building exercise was tackled and how internal communications were developed by the provision of different media.     

A PRIMER ON MATHEMATICAL MODELLING IN ECONOMICS

Abstract The Commission on Graduate Education in Economics had raised several concerns regarding the role of mathematics in graduate training in economics ( Krueger, 1991 ; Colander, 1998, 2005 ). This paper undertakes a detailed scrutiny of the notion of a utility function to motivate and describe the common patterns across mathematical concepts and results that are used by economists. In the process one arrives at a classification of mathematical terms which is used to state mathematical results in economics. The usefulness of the classification scheme is illustrated with the help of a discussion of Arrow's impossibility theorem. Common knowledge of the patterns in mathematical concepts and results could be effective in enhancing communication between students, teachers and researchers specializing in different sub‐fields of economics.